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SPECTRAL ACTION GRAVITY and COSMOLOGICAL MODELS This SPECTRAL ACTION GRAVITY AND COSMOLOGICAL MODELS MATILDE MARCOLLI This paper surveys recent work of the author and collaborators on cosmological models based on the spectral action functional of gravity. A more detailed presentation of the topics surveyed here will be available in the forthcoming book [63]. 1. Spectral Action as a modified gravity model In recent years considerable interest has grown around various modified gravity models and their cosmological implications (for a comprehensive survey, see for instance [23]). These involve a range of different possibilities, including scalar-tensor theories, bigravity, MOND, conformal gravity, f(R) theories, Hoˇrava{Lifschitz gravity, and various braneworld scenarios. Such models have gained importance as a possible source of explanations for dark matter and dark energy phenomena. Observational data can severely constrain modified gravity models (see for example [50]). In this paper we focus on another possible model of modified gravity, which arises naturally in the context of noncommutative geometry, in which gravity is described by the spectral action functional. We review the main aspects of the spectral action model of gravity and the recent development of cosmological applications, outlining where the link to observational constraints can be most significant. The spectral action functional was introduced in [15] as a model of gravity (and gravity coupled to matter) on noncommutative spaces. The generalization to the noncommutative world of a compact Riemannian smooth spin manifold is provided by the notion of a spectral triples (A; H;D), which axiomatizes the relations between the algebra of smooth functions A = C1(X) and the metric on a manifold X, where the metric is encoded in the Dirac operator =D acting on the Hilbert space H = L2(X; S) of square-integrable spinors. The main relation between the algebra and the Dirac operator is expressed by the condition that the commutators [D; a] are bounded operators acting on H. The operator D is required to be self-adjoint and with compact resolvent to encode the analytic properties of the usual Dirac operator on a compact manifold. Under the assumption that the operator D satisfies Tr(jDj−s) < 1 for sufficiently large Re(s), that is, that the spectral triple is finitely summable, the spectral action functional is defined as X (1.1) SΛ;f (D) = Tr(f(D=Λ) = Mult(λ)f(λ/Λ); λ2Spec(D) where f 2 S(R) is an even rapidly decaying function (a smooth approximation to a cutoff function) and Λ 2 R+ is an energy scale parameters that makes D=Λ dimensionless. Thus, one can think of the spectral action as a suitably regularized trace of the Dirac operator. It can be related to the 2 −s heat kernel of D and to the zeta function ζD(s) = Tr(jDj ) via Mellin transform. This provides, in the case where the spectral triple (A; H;D) is an actual compact Riemannian spin manifold, an asymptotic expansion for the spectral action (see [15] and Chapter 1 of CoMa-book) Z X β −β (1.2) SΛ;f (D) ∼Λ!1 fβ Λ −|Dj + f(0) ζD(0); + β2ΣST R 1 β−1 where fβ = 0 f(v) v dv are the momenta of f. The summation is over the points of the non-negative dimension spectrum (poles of the zeta function on the non-negative real line), and 1 2 M.MARCOLLI the coefficients are residues of the zeta function, Z 1 (1.3) −|Dj−β = Res ζ (s); 2 s=β D representing the noncommutative integration in dimension β. The spectral action was proposed in [15] as a possible action functional for gravity coupled to matter, when computed for an almost commutative spectral triple (a product of a manifold and a finite noncommutative space). It was successfully applied to the construction of particle physics models, where its asymptotic expansion reconstructs the Lagrangian of the Standard Model with right handed neutrinos and Majorana masses, [19]. It was also shown in [19] that, in the gravity sector, the asymptotic expansion of the spectral action gives rise to a modified gravity model that includes, in addition to the Einstein{Hilbert action and the cosmological term of General Relativity, also a conformal gravity term (Weyl curvature) and a Gauss{Bonnet gravity term (which is non-dynamical and topological in dimension four). The particle physics models based on the spectral action have recently been shown to accommodate the correct Higgs mass, an additional scalar field (about which more later), supersymmetric models, and Pati{Salaam grand unified theories, see [18], [31], [4], [20]. We will discuss here only the case where the spectral triple (A; H;D) is the commutative spectral triple (C1(X);L2(X; S); =D) associated to a compact spin Riemannian manifold X. In this case the spectral action provides a model of (modified) Euclidean gravity on X (see [46]) which includes, in addition to the usual Einstein{Hilbert action with cosmological constant an additional modified gravity term that includes conformal gravity and Gauss{Bonnet gravity. The reason why the spectral action requires Euclidean signature lies in the property of the Dirac operator: on a compact Riemannian spin manifold the Dirac operator is self-adjoint with compact resolvent, hence in particular the spectrum is discrete and with finite multiplicities, so that (1.1) is well defined, while these properties typically do not hold in the Lorentzian setting. However, though the spectral action itself is defined only in Euclidean signature, it is often possible to make sense of a Wick rotation to Lorentzian signature for the individual terms of its asymptotic expansion. In the case of a 4-dimensional compact Riemannian spin manifold M, the leading terms of the asymptotic expansion of [16] for large Λ correspond to the points β = 0; 2; 4, respectively with contributions 4 2 Tr(f(D=Λ)) ∼ 2Λ f4a0 + 2Λ f2a2 + f0a4: The coefficients a0, a2 and a4 correspond, respectively, to the cosmological term, the Einstein{ Hilbert term, and the Weyl curvature and Gauss{Bonnet modified gravity terms. We will discuss here some of the cosmological implications of this model of gravity. The cosmological implications of conformal gravity models (Weyl curvature) are analyzed for instance in [59] and specific astrophysical and cosmological effects of the presence of the Weyl curvature terms in the spectral action were analyzed in [70] [71], for example with respect to the effects on gravitational wave equations. 2. RGE flows and Early Universe scenarios In the spectral action functional for models of gravity coupled to matter, based on an almost commutative geometry X × F , the choice of the finite noncommutative space F determines the particle physics sector of the model. Indeed, the finite space F = (AF ;HF ;DF ) consists of a spectral triple where the algebra AF and the Hilbert space HF are finite dimensional. The Hilbert space specifies the fermion content of the model, with the representation of AF determining the hypercharges, and the unitaries U(AF ) determine the gauge symmetries. The Dirac operator DF contains the information on the Yukawa parameters (masses and mixing angles) of the particle SPECTRAL ACTION GRAVITY AND COSMOLOGICAL MODELS 3 sector. The gauge boson and the Higgs sector arise from the Dirac operator on the product D = =DX ⊗ 1 + γ5 ⊗ DF , by considering, respectively fluctuations in the manifold direction (gauge bosons) and fluctuations in the finite noncommutative direction (Higgs boson), see Chapter 1 of [24] for a detailed account of this method. In particular, AF = C ⊕ H ⊕ M3(C), with H the quaternions, is the finite algebra that gives rise to an extension of the Minimal Stantard Model with right handed neutrinos and Majorana mass terms, [19]. As shown in [19] (see also Chapter 1 of [24]) in the asymptotic expansion of the spectral action for this almost-commutative geometry, one obtains coefficients of the gravitational term that depend on the Yukawa parameters of the particle sector. This determines certain relations between these parameters at unification energy where the initial conditions for this model are set. More precisely, the asymptotic expansion of the spectral action for the almost commutative geometry is of the form [19] 1 Z p Z p S (D) ∼ R g d4x + γ (Λ) g d4x Λ;f 2κ2(Λ) 0 0 Z Z µνρσp 4 ∗ ∗p 4 + α0(Λ) Cµνρσ C g d x + τ0(Λ) R R g d x 1 Z p Z p + jDHj2 g d4x − µ2(Λ) jHj2 g d4x 2 0 Z Z 2 p 4 4 p 4 − ξ0(Λ) R jHj g d x + λ0(Λ) jHj g d x 1 Z p + (Gi Gµνi + F α F µνα + B Bµν) g d4x; 4 µν µν µν ∗ where G; F; B are the gauge bosons, H is the Higgs field, Cµνρσ is the Weyl curvature, and R R∗ is the (topological) Gauss{Bonnet term. The coefficients in this expansion represent an “effective 2 cosmological constant" γ0(Λ) and an “effective gravitational constant" 8πGeff (Λ) = κ0(Λ). They are given by the expressions 2 1 96f2Λ − f0c(Λ) 3f0 2κ2(Λ) = α0 = − 0 24π2 10π2 2 π b(Λ) 11f0 λ0(Λ) = 2 τ0 = 2 2f0a (Λ) 60π 2 2 f2Λ e(Λ) 1 µ0(Λ) = 2 − ξ0 = 12 f0 a(Λ) 1 f γ (Λ) = (48f Λ4 − f Λ2c(Λ) + 0 d(Λ)) 0 π2 4 2 4 where the terms a; b; c; d; e are functions of the Yukawa parameters Y and the Majorana mass matrix M, which in turn run with the energy scale Λ, y y y y a = Tr(Yν Yν + Ye Ye + 3(Yu Yu + Yd Yd)) y 2 y 2 y 2 y 2 b = Tr((Yν Yν) + (Ye Ye) + 3(Yu Yu) + 3(Yd Yd) ) c = Tr(MM y) d = Tr((MM y)2) y y e = Tr(MM Yν Yν): 4 M.MARCOLLI There are different ways of interpreting this expression in the model.
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